Bounding the smallest singular value of a random matrix without concentration

@article{Koltchinskii2013BoundingTS,
title={Bounding the smallest singular value of a random matrix without concentration},
journal={arXiv: Probability},
year={2013}
}
• Published 12 December 2013
• Mathematics
• arXiv: Probability
Given $X$ a random vector in ${\mathbb{R}}^n$, set $X_1,...,X_N$ to be independent copies of $X$ and let $\Gamma=\frac{1}{\sqrt{N}}\sum_{i=1}^N e_i$ be the matrix whose rows are $\frac{X_1}{\sqrt{N}},\dots, \frac{X_N}{\sqrt{N}}$. We obtain sharp probabilistic lower bounds on the smallest singular value $\lambda_{\min}(\Gamma)$ in a rather general situation, and in particular, under the assumption that $X$ is an isotropic random vector for which $\sup_{t\in S^{n-1}}{\mathbb{E}}| |^{2+\eta} \leq… 144 Citations Sharp lower bounds on the least singular value of a random matrix without the fourth moment condition We obtain non-asymptotic lower bounds on the least singular value of${\mathbf X}_{pn}^\top/\sqrt{n}$, where${\mathbf X}_{pn}$is a$p\times n$random matrix whose columns are independent copies of Sample covariance matrices of heavy-tailed distributions Let$p>2$,$B\geq 1$,$N\geq n$and let$X$be a centered$n$-dimensional random vector with the identity covariance matrix such that$\sup\limits_{a\in S^{n-1}}{\mathrm E}|\langle X,a\rangle|^p\leq
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